The approach described in Finding Multinomial Logistic Regression Coefficients doesn’t provide the best estimate of the regression coefficients. In fact a higher value of LL can be achieved using Solver.
Referring to Figure 2 of Finding Multinomial Logistic Regression Coefficients, set the initial values of the coefficients (range X6:Y8) to zeros and then select Data > Analysis|Solver and fill in the dialog box that appears with the values shown in Figure 1 (see Goal Seeking and Solver for more details) and then click on the Solve button.
Figure 1 – Solver dialog box for Multinomial Logistic Regression
The result is displayed in Figure 2 and 3.
Figure 2 – Multinomial logistic regression model using Solver (part 1)
As you can see the value of LL calculated by Solver is -163.386 (see Figure 3), which is a little larger than the value of -170.269 calculated by the binary model (see Figure 4 of Finding Multinomial Logistic Regression Coefficients).
To test the significance of the coefficients (the equivalent of Figure 5 of Finding Multinomial Logistic Regression Coefficients for the Solver model) we need to calculate the covariance matrix (as described in Property 1 of Finding Multinomial Logistic Regression Coefficients). This is shown in Figure 4.
Figure 4 – Calculation of the Covariance Matrix
The covariance matrix displayed in Figure 4 is calculated using the formulas shown in Figure 5.
Using the results in Figure 2 and 4, we get the result shown in Figure 6.
Figure 6 – Multinomial logistic regression model using Solver (part 3)
The key formulas used to calculate the Cured + Dead table are shown in Figure 7 (the Sick + Dead table is similar).
The forecasted probabilities, based on the multinomial logistic regression model using Solver, of the three outcomes for men and women at a dosages of 24 mg and 24.5 mg is displayed in Figure 8.
Figure 8 – Forecasted probabilities using Solver